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2008-05-12T16:33:41Z

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Vipul: New page: ==Definition== ===Definition with symbols=== Suppose $I,J$ are ideals in a commutative unital ring $R$ . Then the ''product'' of ideals $I$ and
2008-03-16T19:26:50Z

New page: ==Definition== ===Definition with symbols=== Suppose <math>I,J</math> are ideals in a commutative unital ring <math>R</math>. Then the ''product'' of ideals <math>I</math> and <m...

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==Definition==

===Definition with symbols===

Suppose <math>I,J</math> are [[ideal]]s in a [[commutative unital ring]] <math>R</math>. Then the ''product'' of ideals <math>I</math> and <math>J</math>, denoted <math>IJ</math>, is defined in the following equivalent ways:

* It is the additive subgroup generated by all elements of the form <math>ab</math> where <math>a \in I, b \in J</math>
* It is the smallest ideal containing all elements of the form <math>ab</math> where <math>a \in I, b \in J</math>
* It is the ideal defined as the set of elements of the form:

<math>\sum_{i=1}^n a_ib_i</math>

with <math>a_i \in I, b_i \in J</math>

==Facts==

* Product of ideals is commutative and associative. Hence, we can talk of the product of ''more than'' two ideals by simply writing them as a string. The product of ideals <math>I_1, I_2, \ldots, I_n</math>, denoted <math>I_1I_2\ldots I_n</math>, is the subgroup generated by elements of the form <math>a_1a_2\ldots a_n</math> where <math>a_j \in I_j</math> for every <math>j</math>
* We can also use this to define the notion of [[power of an ideal]]. For an ideal <math>I</math>, the ideal <math>I^n</math> is simply <math>II \ldots I</math> written <math>n</math> times. It is the ideal generated by <math>n</math>-fold products of elements from <math>I</math>, and need not be the same as the ideal generated by <math>n^{th}</math> powers of elements from <math>I</math>
* In general, the set of products of elements from <math>I</math> and <math>J</math> is not additively closed. An important exception is the situation where either <math>I</math> or <math>J</math> is a [[principal ideal]].
* The product of two ideals is contained in their intersection, and contains the square of their intersection. In symbols:

<math>(I \cap J)^2 \le IJ \le I \cap J</math>